Every metric space of weight $\lambda=\lambda^{\aleph_0}$ admits a condensation onto a Banach space

TL;DR

The paper proves that metric spaces of certain infinite weights can be bijectively and continuously mapped onto Banach spaces and compact spaces, solving longstanding problems in topology for these classes.

Contribution

It establishes that metric spaces with weight equal to their countable power admit bijective continuous mappings onto Banach and compact spaces, resolving Banach's and Alexandroff's problems in these cases.

Findings

Metric spaces of weight continuum map onto the Hilbert cube.

Spaces with weight λ=λ^{ℵ₀} map onto Hausdorff compact spaces.

Addresses longstanding topological problems for specific infinite weights.

Abstract

In this paper, we have proved that for each cardinal number $\lambda$ such that $\lambda=\lambda^{\aleph_0}$ a metric space of weight $\lambda$ admits a bijective continuous mapping onto a Banach space of weight $\lambda$. Then, we get that every metric space of weight continuum admits a bijective continuous mapping onto the Hilbert cube. This resolves the famous Banach's Problem (when does a metric (possibly Banach) space $X$ admit a bijective continuous mapping onto a compact metric space?) in the class of metric spaces of weight continuum. Also we get that every metric space of weight $\lambda=\lambda^{\aleph_0}$ admits a bijective continuous mapping onto a Hausdorff compact space. This resolves the Alexandroff Problem (when does a Hausdorff space $X$ admit a bijective continuous mapping onto a Hausdorff compact space?) in the class of metric spaces of weight $\lambda=\lambda^{\aleph_0}$.